Robin Spectrum: Two Disks Maximize the Third Eigenvalue

Alexandre Girouard; Richard S. Laugesen

doi:10.1512/iumj.2021.70.8721

Robin Spectrum: Two Disks Maximize the Third Eigenvalue

Research output: Contribution to journal › Article › peer-review

Abstract

The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.

Original language	English (US)
Pages (from-to)	2711-2742
Number of pages	32
Journal	Indiana University Mathematics Journal
Volume	70
Issue number	6
DOIs	https://doi.org/10.1512/iumj.2021.70.8721
State	Published - 2021
Externally published	Yes

Keywords

Neumann
Robin
Steklov
conformal mapping
vibrating membrane

ASJC Scopus subject areas

General Mathematics

Online availability

10.1512/iumj.2021.70.8721

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title = "Robin Spectrum: Two Disks Maximize the Third Eigenvalue",

abstract = "The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.",

keywords = "Neumann, Robin, Steklov, conformal mapping, vibrating membrane",

author = "Alexandre Girouard and Laugesen, {Richard S.}",

note = "Acknowledgements. This research was supported by a grant from the Simons Foundation (no. 429422 to Richard Laugesen). Alexandre Girouard acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC).",

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N1 - Acknowledgements. This research was supported by a grant from the Simons Foundation (no. 429422 to Richard Laugesen). Alexandre Girouard acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC).

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N2 - The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.

AB - The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.

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KW - Robin

KW - Steklov

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KW - vibrating membrane

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Robin Spectrum: Two Disks Maximize the Third Eigenvalue

Abstract

Keywords

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